TITLE: What's between the finite and the infinite?
QUESTION [44 upvotes]: I'm wondering if there are any non-standard theories (built upon ZFC with some axioms weakened or replaced) that make formal sense of hypothetical set-like objects whose "cardinality" is "in between" the finite and the infinite. In a world like that non-finite may not necessarily mean infinite and there might be a "set" with countably infinite "power set".

REPLY [4 votes]: Let me make a few remarks about the constructive aspects. The standard definition is the following: a set $X$ is finite if there is a natural number $n$ and a bijection between $X$ and $\{ i \in \mathbb{N} : i < n \}$. Some of the expected properties are true:

The disjoint union of two finite sets is finite.
The product of two finite sets is finite.
The set of maps between two finite sets is finite.

On the other hand, there are some strange facts:

Subsets of finite sets may not be finite.
Quotients of finite sets may not be finite.

For example, given a proposition $\varphi$, $\{ i \in \mathbb{N} : \varphi \land i < 1  \}$ is finite if and only if $\varphi \lor \lnot \varphi$ holds. (This is because equality in $\mathbb{N}$ is decidable.) Thus one is tempted to look for weaker notions of finiteness.
Here is one alternative. The class of Kuratowski-finite sets is defined inductively as follows:

The empty set is Kuratowski-finite.
Every singleton set is Kuratowski-finite.
The union of two Kuratowski-finite sets is Kuratowski-finite.

It is true that the quotient of a Kuratowski-finite set is automatically Kuratowski-finite. Indeed, every Kuratowski-finite set is in bijection with the quotient of some finite set – thus, one might call them finitely generated sets. In particular, Kuratowski-finiteness is strictly more general than finiteness. On the other hand, subsets of Kuratowski-finite sets may not be Kuratowski-finite.